Abstract The Kawahara equation is a model of capillary-gravity water wave and plasma waves. In this paper, a direct method based on the Jacobi elliptic functions is presented to get analytical solutions of the space-time fractional Kawahara equation. Moreover, new exact solutions are obtained for a nonlinear ordinary differential equation (dF/dξ)2 = PF4(ξ) + QF2(ξ) + R, which appears as an auxiliary equation for solving many partial differential equations. Some notes, theorems, and corollaries are also given for this equation. Solving this auxiliary equation, the solutions of the Duffing equation can also be found. Besides, three test examples are presented to illustrate the application of the method and some solutions are demonstrated by the graphics.

中文翻译：

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时空分数川原方程的新精确解

摘要 Kawahara 方程是毛细管重力水波和等离子体波的模型。本文提出了一种基于 Jacobi 椭圆函数的直接求解时空分数阶 Kawahara 方程的解析解。此外，还获得了非线性常微分方程 (dF/dξ)2 = PF4(ξ) + QF2(ξ) + R 的新精确解，该方程作为求解许多偏微分方程的辅助方程出现。还给出了该方程的一些注释、定理和推论。求解这个辅助方程，也可以找到杜芬方程的解。此外，还给出了三个测试实例来说明该方法的应用，并通过图形演示了一些解决方案。